Device Modeling from Atomistic First Principles: theory of the nonequilibrium vertex correction

page 12012-08-29 NEGF5, Jyvaskyla, Finland

Device Modeling from Atomistic First Principles: theory of the nonequilibrium vertex correction

Eric Zhu1, Leo Liu1, Hong Guo1,2

1 Nanoacademic Technologies Inc. Brossard, QC J4Z 1A7, Canada

2 Dept. of Physics, McGill Univ., Montreal, Quebec, H3A 2T8 Canada

• Introduction: NEGF-DFT;

• 4 critical issues: disorder averaging, band gap, large sizes, verification;

• Two examples: localized doping in Si nanoFET; disorder scattering in MRAM;

• Summary.

Continuum model Atomic model

Goal: simulate a transistor from atomic first principles

(10 nm)3 chunk of Si has ~64,000 atoms.

Picture from Taur and Ning, Fundamentals of Modern VLSI Devices

~100nm~10nm

2012-08-29 page 2NEGF5, Jyvaskyla, Finland

Current: L=22nm Next: L=16nm DFT: ~1,000 atoms

1. Doping & disorder,2. Band gaps,3. Large sizes,4. Accuracy.

Other physics: phonons, magnons, photons, correlations…

… and many other systems with different materials


?Can we calculate ?

In any real device made of any real material, there is a degree of disorder.

Such disorder impacts device operation in serious ways. How do we compute these effects from first principles?

This talk:

Ex 1: Dopant fluctuation gives rise to device-to-device variability


Huge device to device variability.

F.L. Yang et al., in VLSI Technol. Tech. Symp. Dig., pp. 208, June 2007.

If every transistor behaves differently, difficult to design a circuit.


Ex 2: roughness scattering increases resistance of Cu interconnects

With Daniel Gall of RPI.

$: SRC

Ex. 3: disorder effect in topological insulator Bi2Se3

Calculated spin direction

Top surface Bottom surface

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Experiment (Hasan etal) Ab initio (Zhao etal)

Conductance:

Wang, Hu, H.G. PRB 85, 241402 (2012)Zhao, H.G. etal Nano Lett. 11, 2088 (2011).


Quantitative prediction of quantum transport from atomic first principles without any parameter

Semi-empirical device modeling,

10,000 to 100,000 atoms

device parameters

TCADatomic simulations

materials, chemistry, physics

quantum mechanics Physics

device modeling < 5nm (1000 atoms)

New

science engineering

Can we calculate realistic device parameters?


Method - NEGF-DFT: non-equilibrium density matrix

( )r H

DFT

NEGF

NEGF-DFT

R A ~ G dE G G dE

DFT NEGF-DFT `DFT’ in NEGF-DFT is not the usual ground state DFT: density matrix of NEGF-DFT is constructed at non-equilibrium.

No variational solution.

‘DFT’: density functional theory

NEGF: Keldysh nonequilibrium Green’s function

Jeremy Taylor, Hong Guo and Jian Wang, Phys. Rev. B 63, 245407 (2001). M. Brandbyge, J.-L. Mozos, P. Ordejon, J. Taylor, and K. Stokbro, PRB 65, 165401 (2002).

1. Within NEGF-DFT: solving the disorder averaging problem

page 10

2012-08-29 NEGF5, Jyvaskyla, Finland

Doping and disorder scattering from atomic principles

Acknowledgements: Dr. Youqi Ke, Dr. Ke Xia, Dr. Ferdows Zahid, Dr. Eric Zhu, Dr. Lei Liu, Dr. Yibin Hu

Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008); Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010);Ferdows Zahid, Youqi Ke, Daniel Gall and Hong Guo, PRB 81, 045406 (2010); Eric Zhu, Lei Liu and Hong Guo, preprint (2012).

NEGF-DFT/CPA-NVC

Drs. Eric Zhu, Leo Liu, and Yibin Hu: development of the NEGF-DFT/CPA-NVC first principles package Nanodsim (nano-device-simulator) – Nanoacademic Technologis Inc. (www.nanoacademic.com).

A tough problem of atomic calculations: disorder scattering

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NEGF5, Jyvaskyla, Finland

xxBA 1

T. Dejesus, Ph.D thesis, McGill University, 2002.

For any theoretical calculation, disorder averaging must be done.

How to do it in atomistic calculations at non-equilibrium?

Generating many configurations, compute each, and average result very time consuming(Small x, large N)

To build intuition, let’s solve a toy problem exactly

1D tight binding chain

1 2

nearest neighbor coupling

on-site energy

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Self-energies for the leads:

How to handle half-infinite chain ?

Self energy:

The problem is reduced to 3 sites plus self-energies

1 2

L R

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Physical quantities and NEGF

Express physical quantities in terms of NEGF:

The problem is reduced to calculate disorder averaged NEGF

average over disorder configurations

L

R

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Disorder average can be done exactly for the 3-site model

page 15


1 2

L R

Exact solution of the 3-site toy model:

In general, the number of configuration is 2N (N is the number of disorder sites). It is impossible to enumerate and compute all configurations for large N.

We need a better “statistical approach” Coherent potential Approx.

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page 17


CPA - well established formalism

xxBA 1 When there are impurities, translational symmetry is broken. Coherent Potential Approximation (CPA) is an effective medium theory that averages over the disorder and restores the translational symmetry. So, an atomic site has x% chance to be occupied by A, and (1-x)% chance by B.

Rev. Mod. Phys. 46, 466 (1974)

][),( Ra

Lr GGTrVET B. Velicky, Phys. Rev. 183 (1969).

P. Soven, Phys. Rev. 156, 809 (1967).

CPA:

CPA picture: effective media

and are solved from CPA equation

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Implementation:

needs a method that does one atom at a time: LMTO, KKR, etc..

page 19


Non-equilibrium density matrix: nonequilibrium vertex

( )r H

DFT

NEGF

NEGF-DFT

rlAR

AR

AR

GGTr T

dE GG dE G ~

dE GG dE G ~

Average over random disorder: X

Take Home message #1: multiple disorder scattering at non-equilibrium is solved by the non-equilibrium vertex correction theory (NVC) and implemented in NEGF-DFT software Nanodsim.

ARAR G GGG

specular part diffusive part

Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008)

page 20


Essence of Nonequilibrium Vertex Correction (NVC)

X

Conventional vertex correction, i.e. that appears in computing Kubo formula in disordered metal, is done at equilibrium.

NVC is done at non-equilibrium: it is related not only to multiple impurity scattering, but also to the non-equilibrium statistics of the device scattering region.

Implementation: LMTO with atomic sphere approximation, plus CPA and NVC, within NEGF-DFT.

page 21


NVC Equation: some complicated technical details

Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008).

Consistency check: CPA-NVC identity

NVC CPA

CPA

The CPA-NVC identity can also be proved analytically at non-equilibrium: CPA and NVC are consistent approximations (Eric Zhu and H.G., 2012).

The identity is tested numerically: strong check of the code.

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NVC solution for the 3-site toy model

and are solved from NVC equation

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Comparison for the 3-site toy model:

page 24


Excellent ! specular part diffusive part

page 25


Non-toy system:

At equilibrium, fluctuation-dissipation theorem holds.

Left hand side has NVC; right hand side does not.

This gives a very strict check to the NVC formalism as well as to the numerical implementation.

no NVC NVC exact

2. Within NEGF-DFT: solving the band gap problem

page 26


The band gap problem …

Acknowledgements: Dr. Youqi Ke, Dr. Wei Ji, Mathieu Cesar, Dr. Eric Zhu, Dr. Lei Liu,

Dr. Zetian Mi, Dr. Ferdows Zahid

NEGF-DFT-CPA-NVC

The band gap problem of local functionals in DFT

page 27


MBJ computation time is ~LDA

DFT calculation of band gaps:

Some relevant band gaps for transistors materials:

page 28


Some relevant effective masses

page 29


Take home message 2: the band gap problem is practically resolved by MBJ semi-local exchange within LMTO implementation of NEGF-DFT.

page 30


MBJ potential + CPA: works. Good agreement with experimental data.

Experimental data

Calculated data

3. Within NEGF-DFT: solving the large size problem

page 31


Solving large problems from self-consistent first principles.

Acknowledgements:

Dr. Eric Zhu, Dr. Lei Liu (Nanoacademic Technologies Inc.)

Dr. Yibin Hu, Mohammed Harb, Vincent Michaud-Roux (McGill)

J. Maassan, E. Zhu, V. Michaud-Vioux, M. Harb and H.G., to appear in IEEE Proceedings (2012).

Locality: the principle underlying all O(N) methods

Equilibrium density matrix exhibits decaying property:

insulator

metal

Example: Si bulk

LMTO (nanodsim) LCAO (nanodcal)

Locality: the properties of a certain observation region comprising one or a few atoms are only weakly influenced by factors that are spatially far away from this observation region. S. Geodecker Rev. Mod. Phys. (1999)

2012-08-29 page 32


Density matrix computation:

page 33


( )r H

DFT

NEGF

DFT – computes potential and energy levels of the device;

NEGF – non-equilibrium statistics that fills levels;

The self-consistent loop – NEGF-DFT algorithm we use.

Practically, density matrix is divided into two parts: equilibrium and non-equilibrium parts:

no locality locality

R A ~ G dE G G dE

Roadmap for locality-less computation of density matrix of large systems

no preconditioner with preconditioner

qmr SOR ILU H-MatrixJacobigmres bicgstab

algorithms

iterative method direct method

MCS

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Large: ~20,000 atoms

Do not depend on locality

Roadmap (cont.)

single wall double wall thin & long thick & short

algorithms

iterative method direct method

nested dissection principal layer pardiso

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Performance: nano-device simulator (nanodsim)

Nanodsim has been fully parallelized and optimized for both speed and memory costs. The speed is made nearly O(N) along the transport direction.

Benchmark: 160 cores, 3GB / core, 12,800 atomic sites, <30 min/step

speed performance memory performance

Lx = periodic

Ly = 10 nm

Lz = 5, 10 ,15, 20 ,25, 30nm

160 cores

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Solving ~20,000 atomic spheres for open devices at nonequilibrium

Open device structure of Si: parallel NEGF-DFT run on 480 cores.

Structure Size # of atoms NEGF-DFT run Convergence

Leads : 1 X 20 X 1320 atomic spheres

(2880 Orbitals)

Two probe 1 1 X 20 X 206400 atomic spheres

(57600 Orbitals)107 NEGF-DFT steps,

5.5 min/step, total 9.8 hoursPotential 1.0 x 10-5

Charge 1.23 x 10-5 per atom

Two probe 2 1 X 20 X 4012800 atomic spheres


14 min/step, total 46 hours

Potential 1.0 x 10-5

Charge 6.2 x 10-6 per atom

Two probe 3 1 x 20 x 6019200 atomic spheres


30 min/step, total 134 hoursSame as above

Summary: NEGF-DFT modeling has reached realistic device sizes!

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If using tight binding model: huge systems can be done

• Computation time scales linearly with the channel length

• Computation time increases 6~7 times if the cross section doubles

• For Lz = 173.8 nm, 1/3 of computation time is spent on surface Green’s function, and 2/3 spent on transmission calculation

Run on a single computing node with 12 cores and 36 GB memory

1,024,000 Si atoms

10.9 nm ×10.9 nm × 173.8 nm

time = 2 days

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4. How do we know all are well for real devices ?

page 39


Bench-marking the NEGF-DFT atomistic model for device simulations

Acknowledgements:

Lining Zhang (ECE, HKUST), Dr. Ferdows Zahid (Physics, HKU), Dr. Mansun Chan (ECE, HKUST), Dr. Jian Wang (Physics, HKU),

Dr. Jesse Maassen (ECE, Purdue), Dr. Eric Zhu (Nanoacademic).

NEGF-DFT/CPA-NVC

Commercial TCAD tool:

page 40


1328 pages of parameter and physics descriptions

Hundreds of parameters are needed !

page 41


NEGF-DFT/CPA-NVC versus Sentaurus

page 42


Continuum model with external parameters

Sentaurus: Drift-diffusion coupled with Poisson solver in real space grids

Atomic model: parameter-free

NEGF-DFT/CPA-NVC

Potential of Si TFET: p-i-n structure

page 43


Band gaps; doping; disorder; large sizes; computation; …

L. Zhang, F. Zahid, M. Chan, J. Wang, H.G. (2012).

Doping in the channel does not affect the potential profile due to high doping at S/D p-i-n tunnel FET (TFET) potential: almost perfect agreement

Sentaurus (green) versus NEGF-DFT (red)

T=300K

Intrinsic channel8nm

12nm

14nm

Verification for MOSFET channels

page 44


Green: Sentaurus. Red: Nanodsim

New: non-uniform doping – delta doping

page 45


Red – NEGF-DFT

Green – Sentaurus with Fermi statistics

Black – Sentaurus with Boltzman statistics

Atomistic treatment of doping (P-doped)

within CPA formalism

Full double gate FET simulation:

page 46


LG

LS LDoxide

oxideS D

gate

gate

Tox

TSi

Tox

pn+ n+

I-V characteristics

I-V characteristics calculated by atomic model are in good agreement with NanoMos (effective mass model). Atomic model can go much further: surface roughness scattering, inhomogeneous doping, new materials, etc.

Nanodsim (self-consistent atomic) NanoMos (Zhibin Ren’s thesis)

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Example 1: localized doping

page 48


Localized doping suppresses off-state source-to-drain tunneling and reduces performance variability.

Acknowledgement: Dr. Jesse Maassan (ECE, Purdue)

Jesse Maassan & H.G. preprint (2012).

NEGF-DFT-CPA-NVC

New idea: suppressing S-to-D off-state tunneling

page 49


Example 2: MRAM simulations

page 50


Increasing spin transfer torque (STT) by impurity doping.

Acknowledgements:

Dr. Youqi Ke, Prof. Ke Xia, Dr. Eric Zhu, Dr. Dongping Liu, Prof. Xiu Feng Han

Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010)D.P. Liu, X.F. Han and Hong Guo, PRB 85, 245436 (2012).

NEGF-DFT-CPA-NVC

page 51

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MTJ - magnetic tunneling junctions

Picture from W. Butler, Nature Mat., 3, 845 (2004)

TMR =

tot

tottot

I

II

Tunnel barrier is a few atomic layers thick.


Problem: for a given bias, STT is too small, or junction resistance too large.

Spin transfer torque (STT)

Solution: decreasing the junction resistance

page 52


Why resistance is large? Because tunnel barrier is an insulator.

How to reduce resistance? Dope the insulator with metal atoms.

Why it does not work? Because impurity scattering destroys TMR.

Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010)

New idea:

page 53


Can we find a dopant that exponentially decreases resistance, but only linearly decreases TMR?

We thus predict that Zn doping into MgO barrier will solve our problem!

D.P. Liu, X.F. Han and Hong Guo, PRB 85, 245436 (2012).

Newest: CPA to compute variance by evaluating <GGGG>

page 54


Huge device to device variability.

F.L. Yang et al., in VLSI Technol. Tech. Symp. Dig., pp. 208, June 2007.

Eric Zhu & H.G. (2012).

Summary

page 55


By solving 4 critical problems: disorder averaging, band gap, large size and accuracy, NEGF-DFT method can begin to predict device characteristics parameter-free for realistic nanoFET structures.

Other details were included into NEGF-DFT as well: electron-phonon, collinear and non-collinear spin, spin-orbit, photon, high frequency, transient, etc.

Endless application possibilities…

Integration with industry TCAD tools possible.

Further reduction of computation time underway …

page 56


Thank you !Acknowledgements to Canadian funding: NSERC, CIFAR, FQRNT, IRAP, McGill University.

We are grateful to Hong Kong government which funded AoE at HKU where the Sentaurus benchmark was done.

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Device Modeling from Atomistic First Principles: theory of the nonequilibrium vertex correction